Navier Stokes Korteweg Equations Research Articles

Time-Implicit High-Order Accurate Positivity-Preserving Discretizations for the Navier–Stokes and Navier–Stokes–Korteweg Equations

Time-Implicit High-Order Accurate Positivity-Preserving Discretizations for the Navier–Stokes and Navier–Stokes–Korteweg Equations

Large-time behavior of the solutions for an inflow problem of the full compressible Navier–Stokes–Korteweg equations

In this article, we are concerned with the large-time behavior of solutions to an inflow problem for the one-dimensional full compressible Navier–Stokes–Korteweg equations, which can be used to model the compressible viscous fluids with internal capillarity and heat conductivity. Under some suitable assumptions of the far fields and boundary values of the density, the velocity and the absolute temperature, the large-time behavior of the solutions for the initial boundary value problem is govern by the stationary solution and basic waves of the corresponding hyperbolic system. Then the asymptotic stability of the boundary layer, and the superposition of the boundary layer and the three-rarefaction wave are shown provided that the initial perturbation and the strength of the rarefaction wave and stationary wave are small. The proof is mainly based on L2-energy method, some time-decay estimates in Lp-norm for the smoothed rarefaction wave and the space variable decay estimates of the boundary layer. This can be viewed as the first result about the nonlinear stability of the combination of two different wave patterns for the inflow problem of the full compressible Navier–Stokes–Korteweg equations.

Stability of viscous contact wave for the full compressible Navier–Stokes–Korteweg equations with large perturbation

This study focuses on the stability of the viscous contact wave for the one-dimensional full Navier–Stokes–Korteweg equations with density-temperature dependent transport coefficients and large perturbation. Our findings demonstrate a Nishida–Smoller type result, indicating that the solution remains stable under large perturbation as long as γ−1 is sufficiently small. Notably, the smallness of the capillary coefficient is unnecessary. We then employ the initial layer analysis technique to investigate the asymptotic behaviour in the Wk,p norm. We show that the capillary term has a smoothing effect, which implies that the strong solution is indeed a smooth one. Our results represent an improvement over those previously reported in Chen and Sheng (2019 Nonlinearity 32 395–444). Furthermore, by applying the method in this study to the isothermal case, we can achieve a better outcome than Germain and LeFloch (2016 Commun. Pure Appl. Math. 69 3–61).

Global dynamics of large solution for the compressible Navier–Stokes–Korteweg equations

Global dynamics of large solution for the compressible Navier–Stokes–Korteweg equations

Vanishing capillarity–viscosity limit of the incompressible Navier–Stokes–Korteweg equations with slip boundary condition

In this paper, we investigate the vanishing capillarity–viscosity limit of the incompressible Navier–Stokes–Korteweg (NSK) equations in a three-dimensional horizontally periodic strip domain, in which the velocity of the fluid is supplemented with slip boundary condition and the gradient of density with Dirichlet boundary condition on the boundary. We prove that there exists an positive constant T0 independent on the capillarity and viscosity coefficients, such that the incompressible NSK equations have a unique strong solution on [0,T0] and the solution is uniformly bounded in H3. Based on the uniform estimates, we further give the convergence rate in H1 from the solutions of the incompressible NSK equations to the solution of the inhomogeneous incompressible Euler equations as the capillarity and viscosity coefficients go to zero simultaneously.

Interfacial vorticity dynamics for Navier–Stokes–Korteweg system: General theory and application to two-dimensional near-wall cavitation bubble

A theory of interfacial vorticity dynamics is developed for the single-component liquid–vapor flows described by the Navier–Stokes–Korteweg equations. The non-ideal equation of state and surface capillarity are collectively considered via the chemical potential gradient force. Explicit formulas of the interfacial enstrophy and chemical potential fluxes are derived to elucidate their generation mechanisms from the interface, which are expressed by using the fundamental surface physical quantities. Then, the transport equation of the Lamb dilatation (namely, the Lamb vector divergence) is rigorously derived, generalizing the previous results of the Lamb-vector dynamics for single-phase incompressible viscous flows. The derived results are evaluated in a near-wall cavitation bubble combined with the van der Waals equation of state. The roles of surface shear stress and vorticity, surface chemical potential gradient, surface acceleration, surface stretching-shrinking motion and dilatational waves are emphasized in generating the interfacial enstrophy and chemical potential fluxes, featured by the strong vorticity concentration, interface deformation and inward-propagating microjet due to the longitudinal pressure difference. These distinct dynamical features observed near the bubble interface are well captured by the spatial distribution of the Lamb dilatation. Analyzing the dynamics of the Lamb dilatation based on the transport equation demonstrates the significance of the high-curvature rotating surface region and its vicinity in view of their dominance to the unsteady diffusion effect of the Lamb dilatation. The present exposition could provide new physical insights into liquid–vapor flows from the perspective of interfacial vorticity dynamics.

On Rayleigh–Taylor instability in Navier–Stokes–Korteweg equations

This paper focuses on the Rayleigh–Taylor instability in the two-dimensional system of equations of nonhomogeneous incompressible viscous fluids with capillarity effects in a horizontal periodic domain with infinite height. First, we use the modified variational method to construct (linear) unstable solutions for the linearized capillary Rayleigh–Taylor problem. Then, motivated by the Grenier’s idea in (Grenier in Commun. Pure Appl. Math. 53(9):1067–1091, 2000), we further construct approximate solutions with higher-order growing modes to the capillary Rayleigh–Taylor problem and derive the error estimates between both the approximate solutions and nonlinear solutions of the capillary Rayleigh–Taylor problem. Finally, we prove the existence of escape points based on the bootstrap instability method of Hwang–Guo in (Hwang and Guo in Arch. Ration. Mech. Anal. 167(3):235–253, 2003), and thus obtain the nonlinear Rayleigh–Taylor instability result. Our instability result presents that the Rayleigh–Taylor instability can occur in the fluids with capillarity effects for any capillary coefficient kappa >0 if the critical capillary coefficient is infinite. In particular, it improves the previous Zhang’s result in (Zhang in J. Math. Fluid Mech. 24(3):70–23, 2022) with the assumption of smallness of the capillary coefficient.

Energy equality for weak solutions of the compressible Navier–Stokes–Korteweg equations with general pressure law in a bounded domain

In this paper, we study the energy equality for weak solutions to the compressible Navier–Stokes–Korteweg equations with general pressure law in a bounded domain Ω⊂Rd (d=2,3). By exploring the special structure of the nonlinear terms, and using coarea formula, we formulate the sufficient conditions for the regularity of weak solutions to ensure that energy equality holds.

Asymptotic stability of a nonlinear wave for the compressible Navier–Stokes–Korteweg equations in the half space

Asymptotic stability of a nonlinear wave for the compressible Navier–Stokes–Korteweg equations in the half space

Stabilized formulation for phase-transforming flows with special emphasis on cavitation inception

Despite significant advancement in the field of cavitation, the inception of vapor bubbles from pure liquid is still not well understood. In this paper, we propose a numerical algorithm to solve the Navier–Stokes–Korteweg equations, which are phase-field equations that govern the dynamics of multiphase compressible flows. These equations can predict cavitation inception without phenomenological assumptions about mass transfer. We present a modification to the bulk free energy of the fluid that can help us simulate cavitation at much larger length scales than previously possible. We propose a numerical scheme based on a Taylor–Galerkin discretization, a residual-based discontinuity capturing operator and a spatial filtering scheme. Our numerical method can be used for high speed flows with large pressure gradients, which are distinctive features of flows where we observe cavitation inception. We present numerical examples that show the accuracy, stability and robustness of our scheme. Finally, we use the proposed numerical scheme to study benchmark problems in the field of cavitation inception such as flow past a wedge and flow past a bluff body. The proposed numerical scheme opens opportunities to understand different aspects of cavitation in general and inception in particular, in much more detail.

A Structure-Preserving Finite Volume Scheme for a Hyperbolic Reformulation of the Navier–Stokes–Korteweg Equations

In this paper, we present a new explicit second-order accurate structure-preserving finite volume scheme for the first-order hyperbolic reformulation of the Navier–Stokes–Korteweg equations. The model combines the unified Godunov-Peshkov-Romenski model of continuum mechanics with a recently proposed hyperbolic reformulation of the Euler–Korteweg system. The considered PDE system includes an evolution equation for a gradient field that is by construction endowed with a curl-free constraint. The new numerical scheme presented here relies on the use of vertex-based staggered grids and is proven to preserve the curl constraint exactly at the discrete level, up to machine precision. Besides a theoretical proof, we also show evidence of this property via a set of numerical tests, including a stationary droplet, non-condensing bubbles as well as non-stationary Ostwald ripening test cases with several bubbles. We present quantitative and qualitative comparisons of the numerical solution, both, when the new structure-preserving discretization is applied and when it is not. In particular for under-resolved simulations on coarse grids we show that some numerical solutions tend to blow up when the curl-free constraint is not respected.

High friction limits of Euler–Navier–Stokes–Korteweg equations for multicomponent models

High friction limits of Euler–Navier–Stokes–Korteweg equations for multicomponent models

A relaxation model for the non-isothermal Navier-Stokes-Korteweg equations in confined domains

The Navier-Stokes-Korteweg (NSK) system is a classical diffuse interface model which is based on van der Waals' theory of capillarity. Diffuse interface methods have gained much interest to model two-phase flow in porous media. However, for the numerical solution of the NSK equations two major challenges have to be faced. First, an extended numerical stencil is required due to a third-order term in the linear momentum and the total energy equations. In addition, the dispersive contribution in the linear momentum equations prevents the straightforward use of contact angle boundary conditions. Secondly, any real gas equation of state is based on a non-convex Helmholtz free energy potential which may cause the eigenvalues of the Jacobian of the first-order fluxes to become imaginary numbers inside the spinodal region.In this work, a thermodynamically consistent relaxation model is presented which is used to approximate the NSK equations. The model is complimented by thermodynamically consistent non-equilibrium boundary conditions which take contact angle effects into account. Due to the relaxation approach, the contribution of the Korteweg tensor in the linear momentum and total energy equations can be reduced to first-order terms which enables a straightforward implementation of contact angle boundary conditions in a numerical scheme. Moreover, the definition of a modified pressure function enables to formulate first-order fluxes which remain strictly hyperbolic in the entire spinodal region. The present work is a generalization of a previously presented parabolic relaxation model for the isothermal NSK equations.A high-order discontinuous Galerkin spectral element method which supports curved elements and hanging nodes is employed to discretize the system. The relaxation model and its corresponding boundary conditions are validated using solutions of the original NSK model and analytical results for one-, two- and three-dimensional test cases. The simulation of a spinodal decomposition in a three-dimensional porous structure underlines the capability of the presented approach.

Towards a consistent lattice Boltzmann model for two-phase fluids

We revisit the construction of discrete kinetic models for single-component isothermal two-phase flows. Starting from a kinetic model for a non-ideal fluid, we show that, under conventional scaling, the Navier–Stokes equations with a non-ideal equation of state are recovered in the hydrodynamic limit. A scaling based on the smallness of velocity increments is then introduced, which recovers the full Navier–Stokes–Korteweg equations. The proposed model is realized on a standard lattice and validated on a variety of benchmarks. Through a detailed study of thermodynamic properties including co-existence densities, surface tension, Tolman length and sound speed, we show thermodynamic consistency, well-posedness and convergence of the proposed model. Furthermore, hydrodynamic consistency is demonstrated by verification of Galilean invariance of the dissipation rate of shear and normal modes and the study of visco-capillary coupling effects. Finally, the model is validated on dynamic test cases in three dimensions with complex geometries and large density ratios such as drop impact on textured surfaces and mercury drops coalescence.

Boundary vorticity dynamics of two-phase viscous flow

From the Navier–Stokes–Korteweg equations, the exact relations between the fundamental surface physical quantities for the two-phase viscous flow with the diffuse interface are derived, including density gradient, shear stress, vorticity, pressure, enstrophy flux, and surface curvature. These theoretical results provide a solid foundation of the boundary/interfacial vorticity dynamics and a new tool for the analysis of complex interfacial phenomena in two-phase viscous flows. To demonstrate the application of the developed results, simulation of a droplet impacting and spreading on a solid wall is conducted by using a recently developed well-balanced discrete unified gas kinetic scheme, focusing on the spreading process when the separation bubbles form inside the droplet. The distributions of shear stress, pressure, and enstrophy flux at the interface and the wall are analyzed, particularly near the moving contact points and other characteristic points. This example gives an unique perspective to the physics of droplet impingement on a wall.

Asymptotic behavior of solutions to the full compressible Navier–Stokes–Korteweg equations in the half space

The impermeable wall problem of the full compressible Navier–Stokes–Korteweg equations in the half space is investigated in this paper. The asymptotic stability of the rarefaction wave is shown for the small perturbation, provided that the far fields and boundary values satisfy the proper conditions. The proof is completed by the energy method with the help of time-decay estimates for the rarefaction wave.

Zero-Mach limit of the compressible Navier–Stokes–Korteweg equations

We consider the Cauchy problem for the compressible Navier–Stokes–Korteweg system in three dimensions. Under the assumption of the global existence of strong solutions to incompressible Navier–Stokes equations, we demonstrate that the compressible Navier–Stokes–Korteweg system admits a global unique strong solution without smallness restrictions on initial data when the Mach number is sufficiently small. Furthermore, we derive the uniform convergence of strong solutions for compressible Navier–Stokes–Korteweg equations toward those for incompressible Navier–Stokes equations as long as the solution of the limiting system exists.

Large-Time Behavior of Solutions to an Inflow Problem for the Compressible Navier–Stokes–Korteweg Equations in the Half Space

Large-Time Behavior of Solutions to an Inflow Problem for the Compressible Navier–Stokes–Korteweg Equations in the Half Space

Global large solutions and incompressible limit for the compressible Navier–Stokes system with capillarity

Consider the Cauchy problem for the barotropic compressible Navier–Stokes–Korteweg equations in the whole space Rd (d≥2), supplemented with large initial velocity v0 and almost constant initial density ϱ0. In the two-dimensional case, the global solutions are shown in the critical Besov spaces framework without any restrictions on the size of the initial velocity, provided that the pressure admits a stability condition and the volume viscosity is sufficiently large. The result still holds for the higher dimensional case d≥3 under the additional assumption that the classical incompressible Navier-Stokes equations, supplemented with the initial velocity as the Helmholtz projection of v0, admits a global strong solution.

Zero-viscosity-capillarity limit to the planar rarefaction wave for the 2D compressible Navier–Stokes–Korteweg equations

We shall consider the two-dimensional (2D) isentropic Navier–Stokes–Korteweg equations which are used to model compressible fluids with internal capillarity. Formally, the 2D isentropic Navier–Stokes–Korteweg equations converge, as the viscosity and the capillarity vanish, to the corresponding 2D inviscid Euler equations, and we do justify this for the case that the corresponding 2D inviscid Euler equations admit a planar rarefaction wave solution. More precisely, it is proved that there exists a family of smooth solutions for the 2D isentropic compressible Navier–Stokes–Korteweg equations converging to the planar rarefaction wave solution with arbitrary strength for the 2D Euler equations. A uniform convergence rate is obtained in terms of the viscosity coefficient and the capillarity away from the initial time. The key ingredients of our proof are the re-scaling technique and energy estimate, in which we also introduce the hyperbolic wave to recover the physical viscosities and capillarity of the inviscid rarefaction wave profile.